We remark that the classification results
of [1]}, [2]}, [3]} hold locally
in the sense that they relate to a single quadrilateral
cell or a single cube displayed in Figs. REF  and REF .
The important problem of embedding these results into
a two- or three-dimensional lattice, with preservation of the three-dimensional consistency
condition, was discussed in [4]}, [5]}
by using the concept of a Black and White lattice.
One way to solve this problem is
to embed (REF ) into a \(\mathbb {Z}^2\) -lattice with an elementary cell of size greater
than one. In this case, the quad-equation
(REF ) can be extended to a lattice,
and the lattice equation becomes integrable or
linearizable. To this end, following
[1]}, [2]}, [3]},
we reflect the square with respect to the normal
to its right and top sides and then complete a
\(2\times 2\)  lattice by again reflecting one of the
obtained squares in the other direction.
Such procedure is graphically described in Fig. REF .
<FIGURE>